Rev.S08 MAC 1140 Module 7 Exponential Functions

2 Rev.S08 Learning Objectives Upon completing this module, you should be able to 1. perform arithmetic operations on functions. 2. perform composition of functions. 3. calculate inverse operations. 4. identity one-to-one functions. 5. find inverse functions symbolically. 6. use other representations to find inverse functions. 7. distinguish between linear and exponential growth. 8. model data with exponential functions. 9. calculate compound interest. 10. use the natural exponential function in applications. Click link to download other modules.

3 Rev.S08 Exponential Functions Click link to download other modules. 5.1Combining Functions 5.2Inverse Functions and Their Representations 5.3Exponential Functions and Models There are three sections in this module:

4 Rev.S08 Five Ways of Combining Functions If f(x) and g(x) both exist, the sum, difference, product, quotient and composition of two functions f and g are defined by Click link to download other modules.

5 Rev.S08 Example of Addition of Functions Click link to download other modules. Let f(x) = x 2 + 2x and g(x) = 3x - 1Let f(x) = x 2 + 2x and g(x) = 3x - 1 Find the symbolic representation for the function f + g and use this to evaluate (f + g)(2).Find the symbolic representation for the function f + g and use this to evaluate (f + g)(2). x 2 + 2x)3x − 1) (f + g)(x) = (x 2 + 2x) + (3x − 1) x 2 + 5x − 1 (f + g)(x) = x 2 + 5x − ( − 1 = 13 (f + g)(2) = (2) − 1 = 13 or (f + g)(2) = f(2) + g(2)or (f + g)(2) = f(2) + g(2) = (2) + 3(2) − 1 = (2) + 3(2) − 1 = 13 = 13

6 Rev.S08 Example of Subtraction of Functions Click link to download other modules. Let f(x) = x 2 + 2x and g(x) = 3x − 1Let f(x) = x 2 + 2x and g(x) = 3x − 1 −−Find the symbolic representation for the function f − g and use this to evaluate (f − g)(2) −x 2 + 2x)−3x − 1)(f − g)(x) = (x 2 + 2x) − (3x − 1) (f − g)(x) = x 2 − x + 1(f − g)(x) = x 2 − x + 1 So −2 2 − + 1 = 3So (f − g)(2) = 2 2 − = 3

7 Rev.S08 Example of Multiplication of Functions Click link to download other modules. Let f(x) = x 2 + 2x and g(x) = 3x − 1Let f(x) = x 2 + 2x and g(x) = 3x − 1 Find the symbolic representation for the function fg and use this to evaluate (fg)(2) x 2 + 2x)3x − 1)(fg)(x) = (x 2 + 2x)(3x − 1) x 3 + 6x 2 − x 2 − 2x(fg)(x) = 3x 3 + 6x 2 − x 2 − 2x x 3 + 5x 2 − 2x(fg)(x) = 3x 3 + 5x 2 − 2x So 3(2) 3 +5(2) 2 − 2(2) = 40So (fg)(2) = 3(2) 3 +5(2) 2 − 2(2) = 40

8 Rev.S08 Example of Division of Functions Click link to download other modules. Let f(x) = x 2 + 2x and g(x) = 3x − 1Let f(x) = x 2 + 2x and g(x) = 3x − 1 Find the symbolic representation for the function and use this to evaluateFind the symbolic representation for the function and use this to evaluate SoSo

9 Rev.S08 Example of Composition of Functions Click link to download other modules. Let f(x) = x 2 + 2x and g(x) = 3x - 1Let f(x) = x 2 + 2x and g(x) = 3x - 1 Find the symbolic representation for the function f  g and use this to evaluate (f  g)(2)Find the symbolic representation for the function f  g and use this to evaluate (f  g)(2) (f  g)(x) = f(g(x)) = f(3x – 1) = (3x – 1) 2(f  g)(x) = f(g(x)) = f(3x – 1) = (3x – 1) 2 + 2(3x – 1) (f  g)(x) = (3x – 1) ( 3x – 1) + 6x – 2(f  g)(x) = (3x – 1) ( 3x – 1) + 6x – 2 (f  g)(x) = 9x 2 – 3x – 3x x – 2(f  g)(x) = 9x 2 – 3x – 3x x – 2 (f  g)(x) = 9x 2 – 1(f  g)(x) = 9x 2 – 1 So (f  g)(2) = 9(2) 2 – 1 = 35So (f  g)(2) = 9(2) 2 – 1 = 35

10 Rev.S08 How to Evaluate Combining of Functions Numerically? Click link to download other modules. Given numerical representations for f and g in the tableGiven numerical representations for f and g in the table Evaluate combinations of f and g as specified.Evaluate combinations of f and g as specified.

11 Rev.S08 How to Evaluate Combining of Functions Numerically? (Cont.) Click link to download other modules. (f + g)(5) = f(5) + g(5) = = 14 (fg)(5) = f(5)  g(5) = 8  6 = 48 (f  g)(5) = f(g(5)) = f(6) = 7 Try to work out the rest of them now.

12 Rev.S08 How to Evaluate Combining of Functions Numerically? (Cont.) Click link to download other modules. Check your answers:

13 Rev.S08 How to Evaluate Combining of Functions Graphically? Click link to download other modules. Use graph of f and g below to evaluateUse graph of f and g below to evaluate (f + g) (1)(f + g) (1) (f –g) (1)(f – g) (1) (f  g) (1)(f  g) (1) (f/g) (1)(f/g) (1) (f  g) (1)(f  g) (1) y = g(x) y = f(x) Can you identify the two functions? Try to evaluate them now. Hint: Look at the y-value when x = 1.

14 Rev.S08 How to Evaluate Combining of Functions Graphically? Click link to download other modules. y = g(x) y = f(x) (f + g) (1) = f(1) + g(1) = = 3 –––(f – g) (1) = f(1) – g(1) = 3 – 0 = 3 (fg) (1) = f(1)   g(1) = 3   0 = 0 (f/g) (1) is undefined, because division by 0 is undefined. (f  g) (1) = f(g(1)) = f(0) = 2 Check your answer now.

15 Rev.S08 Next, Let’s Look at Inverse Functions and Their Representations. Click link to download other modules.

16 Rev.S08 A Quick Review on Function Click link to download other modules. y = f(x) means that given an input x, there is just one corresponding output y.y = f(x) means that given an input x, there is just one corresponding output y. Graphically, this means that the graph passes the vertical line test.Graphically, this means that the graph passes the vertical line test. Numerically, this means that in a table of values for y = f(x) there are no x- values repeated.Numerically, this means that in a table of values for y = f(x) there are no x- values repeated.

17 Rev.S08 A Quick Example Click link to download other modules. Given y 2 = x, is y = f(x)? That is, is y a function of x?Given y 2 = x, is y = f(x)? That is, is y a function of x? No, because if x = 4, y could be 2 or – 2.No, because if x = 4, y could be 2 or – 2. Note that the graph fails the vertical line test.Note that the graph fails the vertical line test. Note that there is a value of x in the table for which there are two different values of y (that is, x-values are repeated.)Note that there is a value of x in the table for which there are two different values of y (that is, x-values are repeated.) xy 4 – 2 1 –

18 Rev.S08 What is One-to-One? Click link to download other modules. Given a function y = f(x), f is one-to-one means that given an output y there was just one input x which produced that output.Given a function y = f(x), f is one-to-one means that given an output y there was just one input x which produced that output. Graphically, this means that the graph passes the horizontal line test. Every horizontal line intersects the graph at most once.Graphically, this means that the graph passes the horizontal line test. Every horizontal line intersects the graph at most once. Numerically, this means the there are no y- values repeated in a table of values.Numerically, this means the there are no y- values repeated in a table of values.

19 Rev.S08 Example Click link to download other modules. Given y = f(x) = |x|, is f one-to-one?Given y = f(x) = |x|, is f one-to-one? –No, because if y = 2, x could be 2 or – 2. Note that the graph fails the horizontal line test.Note that the graph fails the horizontal line test. Note that there is a value of y in the table for which there are two different values of x (that is, y-values are repeated.)Note that there is a value of y in the table for which there are two different values of x (that is, y-values are repeated.) (2,2) (-2,2) xy – 2 2 –

20 Rev.S08 What is the Definition of a One-to-One Function? Click link to download other modules. A function f is a one-to-one function if, for elements c and d in the domain of f,A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies f(c) ≠ f(d)c ≠ d implies f(c) ≠ f(d) Example: Given y = f(x) = |x|, f is not one-to-one because –2 ≠ 2 yet | –2 | = | 2 |Example: Given y = f(x) = |x|, f is not one-to-one because –2 ≠ 2 yet | –2 | = | 2 |

21 Rev.S08 What is an Inverse Function? Click link to download other modules. f -1 is a symbol for the inverse of the function f, not to be confused with the reciprocal.f -1 is a symbol for the inverse of the function f, not to be confused with the reciprocal. If f -1 (x) does NOT mean 1/ f(x), what does it mean?If f -1 (x) does NOT mean 1/ f(x), what does it mean? y = f -1 (x) means that x = f(y)y = f -1 (x) means that x = f(y) Note that y = f -1 (x) is pronounced “y equals f inverse of x.”Note that y = f -1 (x) is pronounced “y equals f inverse of x.”

22 Rev.S08 Example of an Inverse Function Click link to download other modules. Let F be Fahrenheit temperature and let C be Centigrade temperature.Let F be Fahrenheit temperature and let C be Centigrade temperature. F = f(C) = (9/5)C + 32F = f(C) = (9/5)C + 32 C = f -1 (F) = ?????C = f -1 (F) = ????? The function f multiplies an input C by 9/5 and adds 32.The function f multiplies an input C by 9/5 and adds 32. To undo multiplying by 9/5 and adding 32, one shouldTo undo multiplying by 9/5 and adding 32, one should subtract 32 and divide by 9/5subtract 32 and divide by 9/5 So C = f -1 (F) = (F – 32)/(9/5)So C = f -1 (F) = (F – 32)/(9/5) So C = f -1 (F) = (5/9)(F – 32)So C = f -1 (F) = (5/9)(F – 32)

23 Rev.S08 Example of an Inverse Function (Cont.) Click link to download other modules. F = f(C) = (9/5)C + 32F = f(C) = (9/5)C + 32 C = f -1 (F) = (5/9)(F – 32)C = f -1 (F) = (5/9)(F – 32) Evaluate f(0) and interpret.Evaluate f(0) and interpret. f(0) = (9/5)(0) + 32 = 32f(0) = (9/5)(0) + 32 = 32 When the Centigrade temperature is 0, the Fahrenheit temperature is 32.When the Centigrade temperature is 0, the Fahrenheit temperature is 32. Evaluate f -1 (32) and interpret.Evaluate f -1 (32) and interpret. f -1 (32) = (5/9)( ) = 0f -1 (32) = (5/9)( ) = 0 When the Fahrenheit temperature is 32, the Centigrade temperature is 0.When the Fahrenheit temperature is 32, the Centigrade temperature is 0. Note that f(0) = 32 and f -1 (32) = 0Note that f(0) = 32 and f -1 (32) = 0

24 Rev.S08 Graph of Functions and Their Inverses Click link to download other modules. The graph of f -1 is a reflection of the graph of f across the line y = xThe graph of f -1 is a reflection of the graph of f across the line y = x Note that the domain of f equals the range of f -1 and the range of f equals the domain of f -1.

25 Rev.S08 How to Find Inverse Function Symbolically? Click link to download other modules. Check that f is a one-to-one function. If not, f -1 does not exist.Check that f is a one-to-one function. If not, f -1 does not exist. Solve the equation y = f(x) for x, resulting in the equation x = f -1 (y)Solve the equation y = f(x) for x, resulting in the equation x = f -1 (y) Interchange x and y to obtain y = f -1 (x)Interchange x and y to obtain y = f -1 (x) Example.Example. –f(x) = 3x + 2 – y = 3x + 2 –Solving for x gives: 3x = y – 2 – x = (y – 2)/3 –Interchanging x and y gives: y = (x – 2)/3 –So f -1 (x) = (x – 2)/3

26 Rev.S08 How to Evaluate Inverse Function Numerically? Click link to download other modules. The function is one-to-one, so f -1 exists.The function is one-to-one, so f -1 exists. f -1 (–5) = 1f -1 (–5) = 1 f -1 (–3) = 2f -1 (–3) = 2 f -1 (0) = 3f -1 (0) = 3 f -1 (3) = 4f -1 (3) = 4 f -1 (5) = 5f -1 (5) = 5 xf(x) 1–5 2–

27 Rev.S08 How to Evaluate Inverse Function Graphically? Click link to download other modules. The graph of f below passes the horizontal line test so f is one-to- one.The graph of f below passes the horizontal line test so f is one-to- one. Evaluate f -1 (4).Evaluate f -1 (4). Since the point (2,4) is on the graph of f, the point (4,2) will be on the graph of f -1 and thus f -1 (4) = 2Since the point (2,4) is on the graph of f, the point (4,2) will be on the graph of f -1 and thus f -1 (4) = 2 f(2)=4

28 Rev.S08 What is the Formal Definition of Inverse Functions? Click link to download other modules. Let f be a one-to-one function. Then f -1 is the inverse function of f, if (f -1 o f)(x) = f -1 (f(x)) = x for every x in the domain of f(f -1 o f)(x) = f -1 (f(x)) = x for every x in the domain of f (f o f -1 )(x) = f(f -1 (x)) = x for every x in the domain of f -1(f o f -1 )(x) = f(f -1 (x)) = x for every x in the domain of f -1

29 Rev.S08 Now, Let’s Look at Exponential Functions and Models Click link to download other modules. We will start with our population growth.

30 Rev.S08 Population Growth Click link to download other modules. Suppose a population is 10,000 in January Suppose the population increases by… 500 people per year500 people per year What is the population in Jan 2005?What is the population in Jan 2005? –10, = 10,500 What is the population in Jan 2006?What is the population in Jan 2006? –10, = 11,000 5% per year5% per year What is the population in Jan 2005?What is the population in Jan 2005? 10, (10,000) = 10, = 10,50010, (10,000) = 10, = 10,500 What is the population in Jan 2006?What is the population in Jan 2006? 10, (10,500) = 10, = 11,02510, (10,500) = 10, = 11,025

31 Rev.S08 Population Growth (Cont.) Click link to download other modules. Jan 2005?Jan 2005? 10, = 10,50010, = 10,500 Jan 2006?Jan 2006? 10, (500) = 11,00010, (500) = 11,000 Jan 2007?Jan 2007? 10, (500) = 11,50010, (500) = 11,500 Jan 2008?Jan 2008? 10, (500) = 12,00010, (500) = 12,000 Suppose a population is 10,000 in Jan Suppose the population increases by 500 per year. What is the population in ….

32 Rev.S08 Population Growth (Cont.) Click link to download other modules. Let t be the number of years after Let P(t) be the population in year t. What is the symbolic representation for P(t)? We know…Let t be the number of years after Let P(t) be the population in year t. What is the symbolic representation for P(t)? We know… Population in 2004 = P(0) = 10, (500)Population in 2004 = P(0) = 10, (500) Population in 2005 = P(1) = 10, (500)Population in 2005 = P(1) = 10, (500) Population in 2006 = P(2) = 10, (500)Population in 2006 = P(2) = 10, (500) Population in 2007 = P(3) = 10, (500)Population in 2007 = P(3) = 10, (500) Population t years after 2004 = P(t) = 10,000 + t(500)Population t years after 2004 = P(t) = 10,000 + t(500) Suppose a population is 10,000 in Jan 2004 and increases by 500 per year.

33 Rev.S08 Population Growth (Cont.) Click link to download other modules. Population is 10,000 in 2004; increases by 500 per year P(t) = 10,000 + t(500) P is a linear function of t.P is a linear function of t. What is the slope?What is the slope? 500 people/year500 people/year What is the y-intercept?What is the y-intercept? number of people at time 0 (the year 2004) = 10,000number of people at time 0 (the year 2004) = 10,000 When P increases by a constant number of people per year, P is a linear function of t.

34 Rev.S08 Population Growth (Cont.) Click link to download other modules. Suppose a population is 10,000 in Jan 2004 and increases by 5% per year. Jan 2005?Jan 2005? 10, (10,000) = 10, = 10,50010, (10,000) = 10, = 10,500 Jan 2006?Jan 2006? 10, (10,500) = 10, = 11,02510, (10,500) = 10, = 11,025 Jan 2007?Jan 2007? 11, (11,025) = 11, = 11, , (11,025) = 11, = 11,576.25

35 Rev.S08 Population Growth (Cont.) Click link to download other modules. Suppose a population is 10,000 in Jan 2004 and increases by 5% per year. Let t be the number of years after Let P(t) be the population in year t. What is the symbolic representation for P(t)? We know…Let t be the number of years after Let P(t) be the population in year t. What is the symbolic representation for P(t)? We know… Population in 2004 = P(0) = 10,000Population in 2004 = P(0) = 10,000 Population in 2005 = P(1) = 10, (10,000) = 1.05(10,000) = (10,000) =10,500Population in 2005 = P(1) = 10, (10,000) = 1.05(10,000) = (10,000) =10,500 Population in 2006 = P(2) = 10, (10,500) = 1.05 (10,500) = 1.05 (1.05)(10,000) = (10,000) = 11,025Population in 2006 = P(2) = 10, (10,500) = 1.05 (10,500) = 1.05 (1.05)(10,000) = (10,000) = 11,025 Population t years after 2004 = P(t) = 10,000(1.05) tPopulation t years after 2004 = P(t) = 10,000(1.05) t

36 Rev.S08 Population Growth (Cont.) Click link to download other modules. Population is 10,000 in 2004; increases by 5% per year P(t) = 10,000 (1.05) t P is an EXPONENTIAL function of t. More specifically, an exponential growth function.P is an EXPONENTIAL function of t. More specifically, an exponential growth function. What is the base of the exponential function?What is the base of the exponential function? What is the y-intercept?What is the y-intercept? number of people at time 0 (the year 2004) = 10,000number of people at time 0 (the year 2004) = 10,000 When P increases by a constant percentage per year, P is an exponential function of t.

37 Rev.S08 The Main Difference Between a Linear Growth and an Exponential Growth Click link to download other modules. A Linear Function adds a fixed amount to the previous value of y for each unit increase in xA Linear Function adds a fixed amount to the previous value of y for each unit increase in x For example, in f(x) = 10, x 500 is added to y for each increase of 1 in x.For example, in f(x) = 10, x 500 is added to y for each increase of 1 in x. An Exponential Function multiplies a fixed amount to the previous value of y for each unit increase in x.An Exponential Function multiplies a fixed amount to the previous value of y for each unit increase in x. For example, in f(x) = 10,000 (1.05) x y is multiplied by 1.05 for each increase of 1 in x.For example, in f(x) = 10,000 (1.05) x y is multiplied by 1.05 for each increase of 1 in x.

38 Rev.S08 The Definition of an Exponential Function Click link to download other modules. A function represented by f(x) = Ca x, a > 0, a is not 1, and C > 0 is an exponential function with base a and coefficient C.A function represented by f(x) = Ca x, a > 0, a is not 1, and C > 0 is an exponential function with base a and coefficient C. If a > 1, then f is an exponential growth functionIf a > 1, then f is an exponential growth function If 0 < a < 1, then f is an exponential decay functionIf 0 < a < 1, then f is an exponential decay function

39 Rev.S08 What is the Common Mistake? Click link to download other modules. Don’t confuse f(x) = 2 x with f(x) = x 2Don’t confuse f(x) = 2 x with f(x) = x 2 f(x) = 2 x is an exponential function.f(x) = 2 x is an exponential function. f(x) = x 2 is a polynomial function, specifically a quadratic function.f(x) = x 2 is a polynomial function, specifically a quadratic function. The functions and consequently their graphs are very different.The functions and consequently their graphs are very different. f(x) = 2 x f(x) = x 2

40 Rev.S08 Exponential Growth vs. Decay Click link to download other modules. Example of exponential growth functionExample of exponential growth function f(x) = 3 2 x Example of exponential decay functionExample of exponential decay function Recall, in the exponential function f(x) = Ca xRecall, in the exponential function f(x) = Ca x If a > 1, then f is an exponential growth functionIf a > 1, then f is an exponential growth function If 0 < a < 1, then f is an exponential decay functionIf 0 < a < 1, then f is an exponential decay function

41 Rev.S08 Properties of an Exponential Growth Function Click link to download other modules. Properties of an exponential growth function Domain: (-∞, ∞)Domain: (-∞, ∞) Range: (0, ∞)Range: (0, ∞) f increases on (-∞, ∞)f increases on (-∞, ∞) The negative x-axis is a horizontal asymptote.The negative x-axis is a horizontal asymptote. y-intercept is (0,3).y-intercept is (0,3). ExampleExample f(x) = 3 2 xf(x) = 3 2 x

42 Rev.S08 Properties of an Exponential Decay Function Click link to download other modules. Properties of an exponential decay function Domain: (-∞, ∞)Domain: (-∞, ∞) Range: (0, ∞)Range: (0, ∞) f decreases on (-∞, ∞)f decreases on (-∞, ∞) The positive x-axis is a horizontal asymptote.The positive x-axis is a horizontal asymptote. y-intercept is (0,3).y-intercept is (0,3). ExampleExample

43 Rev.S08 Example of an Exponential Decay: Carbon-14 Dating Click link to download other modules. The time it takes for half of the atoms to decay into a different element is called the half-life of an element undergoing radioactive decay.The time it takes for half of the atoms to decay into a different element is called the half-life of an element undergoing radioactive decay. The half-life of carbon-14 is 5700 years.The half-life of carbon-14 is 5700 years. Suppose C grams of carbon-14 are present at t = 0. Then after 5700 years there will be C/2 grams present.Suppose C grams of carbon-14 are present at t = 0. Then after 5700 years there will be C/2 grams present.

44 Rev.S08 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Click link to download other modules. Let t be the number of years.Let t be the number of years. Let A =f(t) be the amount of carbon-14 present at time t.Let A =f(t) be the amount of carbon-14 present at time t. Let C be the amount of carbon-14 present at t = 0.Let C be the amount of carbon-14 present at t = 0. Then f(0) = C and f(5700) = C/2.Then f(0) = C and f(5700) = C/2. Thus two points of f are (0,C) and (5700, C/2)Thus two points of f are (0,C) and (5700, C/2) Using the point (5700, C/2) and substituting 5700 for t and C/2 for A in A = f(t) = Ca t yields: C/2 = C a 5700Using the point (5700, C/2) and substituting 5700 for t and C/2 for A in A = f(t) = Ca t yields: C/2 = C a 5700 Dividing both sides by C yields: 1/2 = a 5700Dividing both sides by C yields: 1/2 = a 5700

45 Rev.S08 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Click link to download other modules. Half-life

46 Rev.S08 Radioactive Decay (An Exponential Decay Model) Click link to download other modules. If a radioactive sample containing C units has a half- life of k years, then the amount A remaining after x years is given byIf a radioactive sample containing C units has a half- life of k years, then the amount A remaining after x years is given by

47 Rev.S08 Example of Radioactive Decay Click link to download other modules. Radioactive strontium-90 has a half-life of about 28 years and sometimes contaminates the soil. After 50 years, what percentage of a sample of radioactive strontium would remain?Radioactive strontium-90 has a half-life of about 28 years and sometimes contaminates the soil. After 50 years, what percentage of a sample of radioactive strontium would remain? Since C is present initially and after 50 years.29C remains, then 29% remains. Note calculaor keystrokes :

48 Rev.S08 Example of an Exponential Growth: Compound Interest Click link to download other modules. Suppose $10,000 is deposited into an account which pays 5% interest compounded annually. Then the amount A in the account after t years is: A(t) = 10,000 (1.05) tSuppose $10,000 is deposited into an account which pays 5% interest compounded annually. Then the amount A in the account after t years is: A(t) = 10,000 (1.05) t Note the similarity with: Suppose a population is 10,000 in 2004 and increases by 5% per year. Then the population P, t years after 2004 is: P(t) = 10,000 (1.05) tNote the similarity with: Suppose a population is 10,000 in 2004 and increases by 5% per year. Then the population P, t years after 2004 is: P(t) = 10,000 (1.05) t

49 Rev.S08 What is the Compound Interest Formula? Click link to download other modules. If P dollars is deposited in an account paying an annual rate of interest r, compounded (paid) n times per year, then after t years the account will contain A dollars, whereIf P dollars is deposited in an account paying an annual rate of interest r, compounded (paid) n times per year, then after t years the account will contain A dollars, where Frequencies of Compounding (Adding Interest) Frequencies of Compounding (Adding Interest) annually (1 time per year)annually (1 time per year) semiannually (2 times per year)semiannually (2 times per year) quarterly (4 times per year)quarterly (4 times per year) monthly (12 times per year)monthly (12 times per year) daily (365 times per year)daily (365 times per year)

50 Rev.S08 Example Click link to download other modules. Suppose $1000 is deposited into an account yielding 5% interest compounded at the following frequencies. How much money after t years? AnnuallyAnnually SemiannuallySemiannually QuarterlyQuarterly MonthlyMonthly

51 Rev.S08 What is the Natural Exponential Function? Click link to download other modules. The function f, represented byThe function f, represented by f(x) = e x is the natural exponential function where is the natural exponential function where e  e 

52 Rev.S08 An Example of Using Natural Exponential Function in Application (Cont.) Click link to download other modules. Suppose $100 is invested in an account with an interest rate of 8% compounded continuously. How much money will there be in the account after 15 years? A = Pe rt A = $100 e.08(15) A = $332.01

53 Rev.S08 What have we learned? We have learned to 1. perform arithmetic operations on functions. 2. perform composition of functions. 3. calculate inverse operations. 4. identity one-to-one functions. 5. find inverse functions symbolically. 6. use other representations to find inverse functions. 7. distinguish between linear and exponential growth. 8. model data with exponential functions. 9. calculate compound interest. 10. use the natural exponential function in applications. Click link to download other modules.

54 Rev.S08 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Click link to download other modules.